Print version ISSN 0041-6932
Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca June 2009
Igor E. Shparlinski
Abstract. For some natural families of elliptic curves we show that "on average" the exponent of the point group of their reductions modulo a prime grows as .
2000 Mathematics Subject Classification. 11B57, 11G07, 14H52
Key words and phrases. Elliptic curves; Group exponent; Farey fractions.
For integers and such that , we denote by the elliptic curve defined by the affine Weierstraß equation:
For a basic background on elliptic curves, we refer to .
For a prime , we denote by the finite field of elements, which we identify with the set of integers .
When , the set , consisting of the -rational points of together with a point at infinity , forms an abelian group under an appropriate composition rule called addition, and the number of elements in the group satisfies the Hasse bound:
(see, for example, [11, Chapter V, Theorem 1.1]).
It is well-known that is of rank at most two, that is, is isomorphic to
for unique integers and with and . The number is called the exponent of which we denote by . In other words, is the smallest positive such that for all points .
We also put if .
The exponent of elliptic curves has been studied in a number of works, see [4, 7, 8, 9, 10], with a variety of results, each of them indicating that in a "typical case" the exponent tends to be substantially larger than the bound (3) (and its analogue for curves over arbitrary finite fields) guarantees.
W. Duke , among other results, has proved that, assuming the Generalised Riemann Hypothesis, for every fixed integer and with , and arbitrary small , the bound
holds for all but of primes .
holds for all but pairs with ,
In particular, Theorem 1 is nontrivial if
We also show that averaging over gives some additional saving.
the bound (4 ) holds for all but triples with , , .
We note that the condition from , where it is used to simplify the error term, is not neccessary. One can easily extend Theorem 2 for and beyond this range, however since (as in ) small values of and are of main interest we have not done this.
We remark that in  some of the results of  have been extended to hyperelliptic curves. It would also be interesting to obtain analogues of our result for natural families of hyperelliptic curves.
We also consider the set of Farey fractions
For with and two polynomial , the reduction is correctly defined. Various questions concerning the behaviour of the curves on average over and have been studied in . Here we continue to study this family of curves. Certainly the most interesting case is when is small compared to .
is nonzero and the -invariant
is nonconstant. Then for any fixed and all integers and with
holds for all but pairs with , .
The following result follows immediately from the more precise statement of [10, Theorem 3.1].
is at most .
Let and put
where the maximum is taken over all non-principal multiplicative characters modulo such that is the principal character .
Similarly, we define and put
where the maximum is taken over all non-principal multiplicative characters modulo such that is the principal character . For an arbitrary subset , we denote by the number of pairs such that with and . We also denote
The following estimate is given in .
Moreover, it is shown in  that is small "on average" over .
For a prime and an integer with we denote by the number of fractions with and .
It is shown in  that is close to its expected value on average over . More precisely, we have:
Let be the set of pairs for which . Then it is enough to show that
Assume that then by the Burgess bound, see [6, Theorems 12.5 and 12.6], we have . Also, if then have .
Similarly, if then , and if then have .
As before, let be the set of pairs for which . Then it is enough to show that
Let us assume that since the case is similar.
Using the trivial bound for primes , we deduce
Noticing that for the set satisfies the conditions of Lemma 5 , we obtain
By Lemma 4 we have
As before, we use to denote the set of pairs for which .
Let be the set of such that
for some .
Obviously, for any and we have
(since the corresponding curves are isomorphic, see [11, Section III.1]).
We also note that the system of equations
leads to the equation
which has solutions (by the condition on the -invariant ).
Using Lemma 7, we obtain
which concludes the proof.
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Igor E. Shparlinski
Department of Computing, Macquarie University, North Ryde,
Sydney, NSW 2109, Australia
Recibido: 7 de octubre de 2007
Aceptado: 21 de mayo de 2008